Thin mesh overlapping when animated
You can add thickness to parts of your mesh using the below steps:


Switch to edit mode then select the vertices you want to add thickness to
Create a vertex group by pressing the + button on the right of Vertex groups.
Click the Assign button to assign the selected vertices to the newly created group.





Add solidify modifier to your mesh and select the specific vertex group as shown below:





Change the thickness value


This modifier will affect only the vertices that are part of that newly created vertex group.